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Non-ergodicity for C1 expanding maps and g-measures

Published online by Cambridge University Press:  19 September 2008

Anthony N. Quasf
Anthony N. Quasf
Affiliation:
Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB, England
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Abstract

We introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996, pp. 531 - 543
Copyright
Copyright © Cambridge University Press 1996

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